Remark 3 This result actually convolves the transition probability rate, or collision kernel, with a probability density satisfying the condition of the Lower Bound Lemma REF .
It must be noted that this Lower Bound Lemma was not used for the existence theory, it was not needed since transition probability can be pointwise estimated using (REF ) and (REF ). However the reader may have noticed that the existence of solutions relied of the moments Theorem REF  invokes the necessary interpolation estimate (REF ) due to the structure of the Banach space \(L^1_\ell ({\mathbb {R}}^d)\) , with the Lebesgue weight \(\langle v\rangle ^k\) .
As shown in Theorem REF  and later in Section , this path choice produces a solution with
lower upper global bounds for large times, than the one obtained from the classical upper global bound estimates obtained by Bernoulli's equations, as written in [1]} and [2]}. Thus, while this lower bound is not strictly necessary for the argument given in the proof of Theorem REF , it considerably simplifies its proof without sparing the explicit characterization of the exponential rates, which is one of the goals of this review.
However, we found the Lower Bound Lemma necessary for study of \(L^{2}\) -integrability propagation, which allows for the improvement and completeness properties of \(L^p_\ell ({\mathbb {R}}^d)\) , for \(p\in (0,\infty ]\) , propagation proof originally developed in [3]}.

